Phoenix

The phoenix tuning continuum ranges consists of a range of equally-tempered scales ranging from 63.5998 cents (which divides the just 9:5 interval into 16 equal parts), through 63.8141 cents (which divides the just perfect fifth into 11 equal parts, see 11edf). All of these scales stretch the octave by around 8 to 12 cents. A distinctive feature of phoenix-tuned scales is that prime-numbered harmonics are, on average, approximated more reliably than composite ones. Concentrating the error around composites provides greater overall benefit to tempering.

I (Mason Green) chose the name phoenix because these scales approximate most small intervals reasonably well, but they have a noticeable weakness at the 8th harmonic ratio (8:1) which falls almost exactly between two scale degrees; the 9:1, 10:1 and 12:1 are also not approximated well. There is also a long stretch of missed harmonics from 24 (or 25) to 31. Figuratively, the scale, like a phoenix, "dies" at 24 and rises from the ashes again at 31 or 32.

The following harmonics are "split" (not matched): 8, 9, 10, 12, 15, 18. The fact that prime number harmonics are all approximated well, but composites often are not, implies that synthesized tones using the prime harmonic series should make a very good fit with phoenix. The generator of phoenix could be considered an analogue of prime number-generating functions such as Mills' constant.

There is another a very good reason to "split" the eighth harmonic. Having two approximations (one sharp and one flat) for 8 makes it possible to temper out both the syntonic comma (81:80) and the septimal comma (64:63) in the same tuning, if we do some fudging during modulation (for example, by representing the 8:7 with 3 instead of 4 steps, or by using the "blue" octave of 18 steps. The blue octave is not harmonically consonant, but is an interesting melodic interval).

One of the problems with extending beyond 12edo to higher scales is that 12edo tempers out both of these commas, and many familiar melodies and chord progressions in 12edo depend on these comma pumps. Larger scales, with their finer octave divisions, generally temper out out one of these intervals but not both.

By stretching the octave, though, it's possible to have a tuning that extends 12edo while maintaining most of the same equivalencies found in it.

In addition to meantone temperament, phoenix also supports the fenghuang temperament. A scale supports this temperament if it contains a tempered subminor third (which must be closer to 7:6 than to 8:7) that is the octave inversion of a tempered subminor seventh (which must be closer to 7:4 than 12:7). Only stretched-octave temperaments can accomplish this.

Phoenix's approximations to undecimal and tridecimal intervals are dominated by the fact that the 12th harmonic is also split, meaning that 11:12 and 12:13 can each be approximated in two different ways (one corresponding to a whole tone and one to a diatonic semitone).

Fenghuang tempers out the 49:48 diesis. Fenghuang offers unexpected new possibilities for melody and harmony since it violates the usual rule that the octave inversions of sixths and sevenths must always be thirds and seconds, respectively. It is also a good temperament for increasing overall consonance, since 7:6 is more consonant than 8:7, while 7:4 is more consonant than 12:7. This temperament provides a defining aspect of phoenix's sound, particularly in how it handles the 7-limit and in how its "new" intervals (those not shared with 12edo) behave.

12edo can be smoothly deformed into phoenix by gradually moving all pitches toward their nearest neighbors, and finally adding the new "interstitial" pitches at the end.

In phoenix, the pentad 4:5:6:7:8 (represented by scale degrees 0, 6, 11, 15, 19) could be considered the basic chord, along with its utonal counterpart with triads considered incomplete. Unlike tetrads (which sound rather unstable since the top note is a seventh), but like triads, pentads have a great deal of stability and finality due to its highest note being a (tempered) octave, which is even more consonant than the perfect fifth of triads.

The otonal 4:5:6:7:8 has a happy sound, akin to a major triad (which it contains) but richer, whereas the utonal version sounds melancholy, like the minor triad (which it also contains). These are therefore the minor and major chords of phoenix. (Similar pentads can of course be voiced in 19edo as well, but do not sound as good because the sum of squared errors is higher, the 7 in particular is way out of tune in 19edo).

Phoenix scales are best used over only a finite range of around five octaves; harmonics larger than this range are once again matched poorly. A phoenix piano could be built and would have around 97 keys on it (which is only slightly more than an ordinary piano); pianos are especially good instruments for phoenix because they have built-in octave stretching already. Another option though is to end at 87 keys (one fewer than an ordinary piano) since there the largest interval here is the 24th harmonic. Pianos are good instruments for phoenix because they have stretched octaves already.

Interval	Width in steps	16ed9:5 width cents (min)	11edf width cents (max)	Equivalencies (closest listed first)
Chromatic semitone	1		63.814	28:27, 25:24, 33:32
Diatonic semitone	2		127.628	15:14, 16:15
Whole tone	3		191.442	10:9, 9:8, 8:7(functional)
Subminor third	4		255.256	7:6, 8:7
Minor third	5		319.070	6:5
Major third	6		382.884	5:4
Supermajor third	7		446.699	9:7
Perfect fourth	8		510.513	4:3
Augmented fourth	9		574.327	7:5
Diminished fifth	10		638.141
Perfect fifth	11		701.955	3:2
Subminor sixth	12		765.769	14:9
Minor sixth	13		829.583	8:5
Major sixth	14		893.397	5:3
Subminor seventh	15		957.211	7:4
Minor seventh	16		1021.025	9:5
Major seventh	17		1084.840	15:8
Diminished octave, blue octave	18		1148.654	11:6
Octave (stretched)	19		1212.468	2:1
Chromatic minor ninth	20		1276.282
Diatonic minor ninth	21		1340.096	13:6
Major ninth	22		1403.910	9:4
Subminor tenth	23		1467.724	7:3
Minor tenth	24		1531.538	17:7, 12:5
Major tenth	25		1595.352	5:2, 19:8
Minor eleventh	26		1659.167	13:5
Perfect eleventh	27		1722.980	8:3
Augmented eleventh, diminished twelfth	28		1786.795	14:15
Minor twelfth	29		1850.609
Perfect twelfth, tritave, 3rd harmonic	30		1914.423	3:1
Minor	31		1978.237
Neutral thirteenth	32		2042.051	13:4
Major thirteenth	33		2105.865	10:3, 27:8
Subminor fourteenth	34		2169.680	7:2
Minor fourteenth	35		2233.493	18:5, 11:3
Major fourteenth	36		2297.307	15:4
Double octave, fifteenth (flat)	37		2361.121
Double octave, fifteenth (sharp)	38		2424.935	4:1
Minor sixteenth	39		2488.750	19:9
Neutral sixteenth	40		2552.564	13:4
Major sixteenth	41		2616.378	9:2
Minor seventeenth	42		2680.192	14:3
Neutral seventeenth	43		2744.006
Major seventeenth; fifth harmonic	44		2807.82	5:1
Minor eighteenth	45		2871.634	21:4
Major eighteenth	46		2935.448	11:2 (flat)
Augmented eighteenth; diminished nineteenth	47		2999.262	11:2 (sharp)
Minor nineteenth	48		3063.076
Perfect nineteenth, sixth harmonic	49		3126.890	6:1
Minor twentieth	50		3190.705	19:3
Neutral twentieth	51		3254.519	13:2
Major twentieth	52		3318.333	27:4
(Sub)minor twenty-first; 7th harmonic	53		3382.147	7:1
Minor/neutral twenty-first	54		3445.961	22:3
Major twenty-first	55		3509.775	15:2
Twenty-second, triple octave, 8th harmonic (flat)	56		3573.589	8:1 (flat)
Twenty-second, triple octave, 8th harmonic (sharp)	57		3637.403	8:1 (sharp)
Minor twenty-third	58		3701.217	17:2
Neutral twenty-third	59		3765.031
Major twenty-third; 9th harmonic	60		3828.845	9:1
Minor twenty-fourth	61		3892.659	19:2
Neutral twenty-fourth	62		3956.474	29:3
Major twenty-fourth, decade	63		4020.288	10:1 (sharp), 31:3 (flat)
Twenty-fifth	64		4084.102	21:2
Half-augmented twenty-fifth; 11th harmonic	65		4147.916	11:1
Diminished twenty-sixth	66		4211.730	34:3
Minor twenty-sixth	67		4275.544	12:1 (flat)
Major twenty-sixth	68		4339.358	12:1 (sharp)
Minor twenty-seventh	69		4403.172
Neutral twenty-seventh, thirteenth harmonic	70		4466.986	13:1
Major twenty-seventh	71		4530.800	27:2
Minor twenty-eighth, 14th harmonic	72		4594.615	14:1
Neutral twenty-eighth	73		4658.429	15:1 (flat)
Major twenty-eighth	74		4722.243	15:1 (sharp)
Twenty-ninth; quadruple octave, 16th harmonic	75		4786.057	16:1
Half-augmented twenty-ninth	76		4849.871	33:2
Minor thirtieth, 17th harmonic	77		4913.685	17:1
Chromatic major thirtieth	78		4977.499	35:2
Diatonic major thirtieth	79		5041.313	37:2
Minor thirty-first, 19th harmonic	80		5105.127	19:1
Neutral thirty-first	81		5168.941
Major thirty-first	82		5232.755	41:2
Thirty-second, 21st harmonic	83		5296.570	21:1
Half-augmented thirty-second, 22nd harmonic	84		5360.384	22:1
23rd harmonic	85		5424.198	23:1
Thirty-third	86			24:1, 47:2 *
Half-augmented thirty-third	87		5551.826	49:2
Minor thirty-fourth	88		5615.640	51:2
Major thirty-fourth	89		5679.454	53:2
Supermajor thirty-fourth	90		5743.268	55:2
Minor thirty-fifth	91		5807.082	57:2
Neutral/major thirty-fifth	92		5870.896
31st harmonic	93		5934.710	31:1 (wide-end only)
32nd harmonic	94		5998.52	32:1
33rd harmonic	95		6062.339	33:1
34th harmonic	96		6126.153	34:1 (narrow-end only)

Harmonics beyond the 34th are not matched well.

The wide end of the phoenix continuum matches 86 scale degrees to the 24th harmonic, whereas the narrow end matches 86 degrees to 47:2. The wide end of the continuum matches the 31st harmonic, the narrow end does not. On the other hand, the narrow end of the continuum matches the 34th harmonic, unlike the wide end.